The Involutions of Compact Symmetric Spaces, V

Abstract

In this part, we will prove that the roots of a symmetric space $M$ defined with the curvature (by using the Jacobi equation) make a root system $R(M)$ and determine their multiplic- ities. Thus we reestablish the known facts in a more geometric way (including the fact that the roots of a simple Lie algebra make a root system). Conversely, $R^{m}(M),$ $R(M)$ with the information of the multiplicity, allows one to recover the curvature of $M$ ; that is, $R^{m}(M)$ is a simple and complete description of the curvature tensor (although we will not give a prool).